Fractional Volterra-type operator induced by radial weight acting on Hardy space

Given a radial doubling weight $\mu$ on the unit disc $\mathbb{D}$ of the complex plane and its odd moments ${\mu }_{2n+1}={\int }_0^1{s}^{2n+1}\mu (s)ds$, we consider the fractional derivative $$D^{\mu }(f)(z)=\sum _{n=0}^{\mathrm{\infty }}\frac{\widehat{f}(n)}{{\mu }_{2n+1}}{z}^n,$$ of a function $f(z)={\sum }_{n=0}^{\mathrm{\infty }}\widehat{f}(n)z^n$ analytic in $\mathbb{D}$. We also consider the fractional integral operator $I^{\mu }(f)(z)={\sum }_{n=0}^{\mathrm{\infty }}{\mu }_{2n+1}\widehat{f}(n)z^n$, and the fractional Volterra-type operator $$V_{\mu ,g}(f)(z)=I^{\mu }(f\cdot D^{\mu }(g))(z),f\in \mathcal{H}(\mathbb{D}),$$ for any fixed $g\in \mathcal{H}(\mathbb{D})$. We prove that $V_{\mu ,g}$ is bounded (compact) on a Hardy space $H^p$, $0<p<\mathrm{\infty }$, if and only if $g$ belongs to $\text{BMOA}$ ($\text{VMOA}$). Moreover, if ${\int }_0^1\frac{{({\int }_r^1\mu (s)ds)}^p}{(1-r{)}^2}dr=+\mathrm{\infty }$, we prove that $V_{\mu ,g}$ belongs to the Schatten class $S_p(H^2)$ if and only if $g=0$. On the other hand, if $\frac{{({\int }_r^1\mu (s)ds)}^p}{(1-r{)}^2}$ is a radial doubling weight it is proved that $V_{\mu ,g}\in S_p(H^2)$ if and only if $g$ belongs to the Besov space $B_p$. En route, we obtain descriptions of $H^p$, $\text{BMOA}$, $\text{VMOA}$ and $B_p$ in terms of the fractional derivative $D^{\mu }$.